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4.3 Derivability and Continuity of A Function At A Point

THEOREM: If a function ‘f ’ is derivable at x, then ‘f ’ is continuous at x.

PROOF: We are given that ‘f’ is derivable at x

\ f ’ (x) =

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REMARK: The converse of this theorem i.e. " If ‘f ’ is continuous at ‘x’ , then ‘f’ is derivable at ‘x’ " is not true. This can be seen from the following examples.

Example 3

Shows that the function f defined by f(x) = |x| is continuous but not derivable at x = 0.

We have f (0) = 0 and |x| = 0. Hence f (x) = f (0) and ‘f ’ is continuous at x = 0.

From Example 1 we have already proved that ‘f ’ is not derivable at x = 0

Example 4

If f (x) = x sin , x ¹ 0 and f (0) = 0, show that ‘f ’ is continuous but not derivable at x = 0.

We have f (0) = 0 and x sin = (a finite limit) ´ 0 = 0 since sin always lies between -1 and 1 for any value of x.
Therefore -|x|
£ x sin£ |x|

But    |x| = 0 and - |x| = 0 Þ x sin = 0

Now  f "(0) =

= , for the given function.

\ f ’ (0) = which does not exist, Hence ‘f ’ is not derivable at x = 0

Index

4. 1 Derivability At A Point
4. 2 Derivability In An Interval
4. 3 Derivability And Continuity Of A Function At A Point
4. 4 Some Counter Examples
4. 5 Interpretation Of Derivatives
4. 6 Theorems On Derivatives (differentiation Rules)
4. 7 Derivatives Of Standard Functions
4. 8 Derivative Of A Composite Function
4. 9 Differentiation Of Implicit Functions
4.10 Derivative Of An Inverse Function
4.11 Derivatives Of Inverse Trigonometric Functions
4.12 Derivatives Of Exponential & Logarithmic Functions
4.13 Logarithmic Differentiation
4.14 Derivatives Of Functions In Parametric Form
4.15 Higher order Derivatives

Chapter 5